**1. Introduction**

Three-phase asynchronous motors are commonly used in industrial and agricultural production due to their advantages of simple structure, ruggedness, and low price [1–4]. The emergence of vector control allows the AC speed regulation system to have good speed regulation performance, just like a DC speed regulation system [5]. However, vector control has disadvantages, such as dependence on accurate mathematical models, poor adaptability to instruction changes, and sensitivity to changes in the system parameters. Even if the motor parameters and rotor flux linkage are known accurately, decoupling can be achieved under steady-state conditions, and there are still couplings during the field-weakening speed regulation. The nonlinearity of the magnetization curve of the ferromagnetic material in the motor [6] leads to the nonlinearity of the motor inductance, and the change of the inductance parameter reduces the speed control effect of the vector control. The AC control system has the characteristics of non-linearity [1,4], strong couplings, and having multiple variables, which means the traditional control method based on a precise mathematical model faces severe challenges.

The active disturbance rejection control (ADRC) theory was proposed for the control of nonlinear uncertain systems. The system is linearized by compensating for the observed total disturbance. The compensated system can be converted into an integrator series independent of whether the object is deterministic, linear, or nonlinear, or whether it is time-varying or time-invariant. At present, the ADRC theory has been applied to a six-rotor aircraft [7], a permanent magne<sup>t</sup> synchronous motor [8], DC–DC boost converter [9], and other fields. Abdul-Adheem et al. [10] applied the improved ADRC to the decoupling control of multivariable systems. The coupling is divided into two parts: static coupling (control input of the system) and dynamic coupling (parts other than the control input to the system). All ADRC pass a reversible static matrix (approximately reversible also applies) that is used for decoupling. Compared with the traditional decentralized control method (an automatic disturbance rejection controller designed independently for each part of the system), the improved ADRC decoupling algorithm uses part of the system model information, easing the burden of the extended state observer (ESO) such that the decoupling e ffect is better. Although both numerical simulations and physical verification show that the ADRC controller has a good control e ffect, the large number of modeled non-linear links between the ADRC require high system hardware requirements and increases the di fficulty of real-time control. The second-order auto-disturbance rejection controller has 15 parameters that need to be adjusted, and the direction of the parameter adjustment is di fficult to determine, which brings certain di fficulties in the practical application of the controller. In short, many factors limit the popularity and engineering application of ADRC.

American scholar Gao Zhiqiang explored the connotation and meaning of the idea of auto-disturbance control. He was inspired by the concept of "time scale" [11] proposed by Han Jingqing researchers and proposed the concept of "frequency scale." The parameter setting is carried out through the pole configuration in the frequency domain, and the parameters to be set are reduced to three, which greatly promotes the development and application of the auto-disturbance control theory. Since then, the linear auto-rejection controller has been used in fault detection [12], a wind energy conversion system [13], maximum power point tracking [14], and other fields. Li et al. [15] adopted the concept of "relative order" to determine the order of the linear ADRC (LADRC) controller and designed a second-order LADRC controller to suppress the harmonics to the grid in the AC microgrid. Laghridat et al. [13] applied LADRC to the control of generators and grid-side converters. Compared with the ADRC, an LADRC has the advantages of a fixed structure, an independent object model, clear physical meaning, easy theoretical analysis [13,16], and easy engineering application. However, in practical applications, it is found that the anti-interference performance of LADRC decreases rapidly with the increase of interference and input frequency, which is related to the insu fficient performance of a traditional ESO [17].

This study took the three-phase cage asynchronous motor as the control object, established a mathematical model for it according to the rotor flux linkage orientation, and introduced the structure of the ADRC controller and the role of each component. First, the structure and function of each part of the ADRC controller are introduced. Then, based on the basic principle of deviation control, the adjustment process of each state variable of traditional LESO is discussed and improved. Through theoretical analysis, the stability proof and precision analysis of an improved LESO are given. From the frequency domain, the convergence, tracking, and immunity of the improved linear auto-disturbance controller are analyzed. In the time domain, a large deviation band of the initial value of the internal state variable of the observer is seen and the corresponding value of the system's overshooting is compared. Finally, the control e ffects of the two controllers are compared based on results from Matlab/Simulink digital simulation software (Developed by MathWorks in Natick, MA, USA, and dealt by MathWorks Software (Beijing) Co., Ltd. in Beijing, China).

### **2. Mathematical Modeling of an Asynchronous Motor and Introduction to Classic LADRC**

### *2.1. Mathematical Modeling of the Rotor Flux Orientation of Induction Motor*

To facilitate the research, it was necessary to treat the motor as an ideal motor; therefore, it was necessary to make the following assumptions [18]:


The three-phase winding voltage balance equation in the three-phase stationary coordinate system is shown in Figure 1:

**Figure 1.** Physical model of an asynchronous motor.

The three-phase winding voltage balance equation in the three-phase stationary coordinate system is shown in Equation (1):

$$
\begin{bmatrix} u\_A \\ u\_B \\ u\_C \\ u\_d \\ u\_b \\ u\_c \\ u\_c \end{bmatrix} = \begin{bmatrix} R\_s & 0 & 0 & 0 & 0 & 0 \\ 0 & R\_c & 0 & 0 & 0 & 0 \\ 0 & 0 & R\_s & 0 & 0 & 0 \\ 0 & 0 & 0 & R\_r & 0 & 0 \\ 0 & 0 & 0 & 0 & R\_r & 0 \\ 0 & 0 & 0 & 0 & 0 & R\_r \end{bmatrix} \begin{bmatrix} i\_A \\ i\_B \\ i\_C \\ i\_B \\ i\_B \\ i\_c \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \psi\_A \\ \psi\_B \\ \psi\_C \\ \psi\_B \\ \psi\_B \\ \psi\_C \end{bmatrix} \tag{1}
$$

where *uA*,*uB*,*uC*,*ua*,*ub* and *uc* are the instantaneous values of the stator (A, B, C) and rotor phase (a, b, c) voltages; *iA*,*iB*,*iC*,*ia* ,*ib* , and *ic* are the instantaneous values of the stator (A, B, C) and rotor (a, b, c) phase currents; and ψ*<sup>A</sup>*,ψ*<sup>B</sup>*,ψ*<sup>C</sup>*,ψ*<sup>a</sup>*,ψ*<sup>b</sup>*,ψ*<sup>c</sup>* are the full flux of each phase winding. *Rs* and *Rr* are the resistances of the stator and rotor windings, respectively.

The flux of each winding of an asynchronous motor is the sum of its own self induction flux and the mutual inductance flux of other windings. Therefore, the flux of the six windings can be expressed as follows: 

$$
\begin{bmatrix}
\psi\_s \\
\psi\_r
\end{bmatrix} = \begin{bmatrix}
L\_{ss} & L\_{sr} \\
L\_{rs} & L\_{rr}
\end{bmatrix} \begin{bmatrix}
i\_s \\
i\_r
\end{bmatrix} \tag{2}
$$

where ψ*s* = + ψ*A* ψ*B* ψ*C* ,*<sup>T</sup>*,ψ*<sup>r</sup>* = + ψ*a* ψ*b* ψ*c* ,*T*,*is* = + *iA iB iC* ,*T*, and *ir* = + *ia ib ic* ,*T*. The stator inductance matrix is:

$$\mathbf{L}\_{\rm ss} = \begin{bmatrix} L\_{\rm ms} + L\_{\rm ls} & -\frac{1}{2}L\_{\rm ms} & -\frac{1}{2}L\_{\rm ms} \\ -\frac{1}{2}L\_{\rm ms} & L\_{\rm ms} + L\_{\rm ls} & -\frac{1}{2}L\_{\rm ms} \\ -\frac{1}{2}L\_{\rm ms} & -\frac{1}{2}L\_{\rm ms} & L\_{\rm ms} + L\_{\rm ls} \end{bmatrix} . \tag{3}$$

The rotor inductance matrix is:

$$\mathbf{L}\_{rr} = \begin{bmatrix} L\_{\rm wss} + L\_{\rm lr} & -\frac{1}{2}L\_{\rm sws} & -\frac{1}{2}L\_{\rm sws} \\ -\frac{1}{2}L\_{\rm ms} & L\_{\rm ms} + L\_{\rm lr} & -\frac{1}{2}L\_{\rm ms} \\ -\frac{1}{2}L\_{\rm ms} & -\frac{1}{2}L\_{\rm sws} & L\_{\rm ms} + L\_{\rm lr} \end{bmatrix}. \tag{4}$$

The mutual inductance matrix of the stator and rotor is:

$$\mathbf{L}\_{rs} = \mathbf{L}\_{sr}^T = L\_{ms} \begin{bmatrix} \cos\theta & \cos(\theta - \frac{2\pi}{3}) & \cos(\theta + \frac{2\pi}{3}) \\ \cos(\theta + \frac{2\pi}{3}) & \cos\theta & \cos(\theta - \frac{2\pi}{3}) \\ \cos(\theta - \frac{2\pi}{3}) & \cos(\theta + \frac{2\pi}{3}) & \cos\theta \end{bmatrix} . \tag{5}$$

The voltage equation of the asynchronous motor in the rotating orthogonal coordinate system can be obtained from a Park transform:

$$
\begin{bmatrix} u\_{sd} \\ u\_{sq} \\ u\_{rd} \\ u\_{rq} \end{bmatrix} = \begin{bmatrix} R\_s & 0 & 0 & 0 \\ 0 & R\_s & 0 & 0 \\ 0 & 0 & R\_r & 0 \\ 0 & 0 & 0 & R\_r \end{bmatrix} \begin{bmatrix} i\_{sd} \\ i\_{sq} \\ i\_{rd} \\ i\_{rq} \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \psi\_{sd} \\ \psi\_{sq} \\ \psi\_{rd} \\ \psi\_{rq} \end{bmatrix} + \begin{bmatrix} -\omega\_1 \psi\_{sq} \\ \omega\_1 \psi\_{sd} \\ -(\omega\_1 - \omega) \psi\_{rq} \\ (\omega\_1 - \omega) \psi\_{rd} \end{bmatrix} \tag{6}
$$

where *usd*, *usq*, *urd*, and *urq* are the components of the voltage of the stator and rotor sides along the d and q axes; *isd*, *isq*, *ird*, *irq* are the components of the current of the stator and rotor sides along the d and q axes; ψ*sd*, ψ*sq*, ψ*rd*, and ψ*rq* are the components of the magnetic flux of the stator and rotor sides along the *d* and *q* axes; and ω1 and ω are the synchronous angular velocity and the rotor angular velocity, respectively.

The flux equation of the asynchronous motor in the synchronous rotation orthogonal coordinates system is:

$$
\begin{bmatrix}
\psi\_{sd} \\
\psi\_{sq} \\
\psi\_{rd} \\
\psi\_{rq}
\end{bmatrix} = \begin{bmatrix}
L\_s & 0 & L\_m & 0 \\
0 & L\_s & 0 & L\_m \\
L\_m & 0 & L\_r & 0 \\
0 & L\_m & 0 & L\_r
\end{bmatrix} \begin{bmatrix}
i\_{sd} \\
i\_{sq} \\
i\_{rd} \\
i\_{rq}
\end{bmatrix}' \tag{7}
$$

where *Ls*, *Lr*, and *Lm* are stator inductance, rotor inductance, and mutual inductance between the stator and rotor, respectively. ψ*rd* = ψ*r* when the rotor flux is oriented as in Figure 2.

**Figure 2.** Rotor flux orientation.

Selecting the stator currents and rotor flux as the state variables, and the stator voltages as the control variables, a fourth-order simplified nonlinear differential equation can be given as follows:

$$\begin{cases} \dot{\omega}\_{r} = k\_{1}\psi\_{r}i\_{sq} - \frac{n\_{p}}{f}T\_{L\_{r}}\\ \dot{\psi}\_{rd} = \frac{L\_{m}}{T\_{r}}i\_{sd} - \frac{1}{T\_{r}}\psi\_{r},\\ \dot{i}\_{sd} = -k\_{2}i\_{sd} + k\_{3}\psi\_{r} + \omega\_{1}i\_{sq} + \frac{1}{\sigma L\_{s}}u\_{sd},\\ \dot{i}\_{sq} = -k\_{2}i\_{sq} - \frac{L\_{m}}{\sigma L\_{r}L\_{r}}\psi\_{r}\omega\_{r} - \omega\_{1}i\_{sd} + \frac{1}{\sigma L\_{s}}u\_{sq}. \end{cases} \tag{8}$$

In the above state equations, *np*, *Tr*, and *TL* represent the number of pole pairs of the motor, the load torque, and the rotor time constant, respectively. The coupling between the state variables in Equation (8) causes the nonlinearity of the system [4,10]. The vector control block diagram of the asynchronous motor is shown in Figure 3, in which automatic speed regulator (*ASR*), the automatic current torque regulator (*ACTR*), and the automatic current magnetic regulator (*ACMR*) are the controllers of the vector speed regulation system.

**Figure 3.** Vector control structure of an asynchronous motor. ASR: automatic speed regulator, ACTR: Automatic torque current regulator, ACMR: Automatic magnetic field current regulator SVPWM: Space Vector Pulse Width Modulation.

In Equation (8), there is a cross-coupling term in the state equation of the asynchronous motor, thus resulting in the mutual effect of control of the torque component and the excitation component of the stator current, which further affects the dynamic and static performance of the system. The graphical representation of the coupling term is shown in Figure 4. Decoupling can be achieved if the coupling terms are observed and compensated for [19].

**Figure 4.** Coupling structure of an asynchronous motor.

### *2.2. LADRC Introduction and First-Order LADRC Design*

The ADRC consists of a tracking differentiator (TD) [20,21], an extended state observer (ESO) [22,23], and a nonlinear states error feedback control laws (NLSEF) . Among them, the TD can arrange the transition process for the given input, reduce the "impact" of the original error on the system caused by the given mutation, and realize the differential signal extraction; ESO can observe the state variables and total disturbances (unmodeled dynamics and external disturbances) of the controlled object in a real-time manner; and the NLSEF is used to improve the dynamic characteristics of the closed-loop system. The linear ADRC only replaces the nonlinear part of the original controller with the linear part, where the structure of the controller is shown in Figure 5.

**Figure 5.** Structure of an active disturbance rejection controller (TD: tracking differentiator, NLSEF: nonlinear state error feedback, ESO: extended state observer).

Consider the first-order system [24]:

$$
\dot{y} = -ay + w + bu\_\prime \tag{9}
$$

where *u* is the output of the controller, *y* is the system output, *w* is the external disturbance, *a* is the system parameter, *b* is the controller gain that satisfies *b* ≈ *b*0; the parameters *a* and *b* are unknown. Let *x*1 = *y* and *x*2 = *f*(*y*, *w*) = −*ay* + *w* + (*b* − *b*0)*<sup>u</sup>*, where *x*1 represents the system output and *x*2 represents the total disturbance of the system. Assuming that *f*(*y*, *w*) is derivable and satisfies . *f*(*y*, *w*) = *h*, the state variables *x*1 = *y* and *x*2 are selected to establish the continuous extended state equation s shown in Equation (10):

$$\begin{cases} \begin{bmatrix} \dot{\mathbf{x}}\_1\\ \dot{\mathbf{x}}\_2 \end{bmatrix} = \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix} \begin{bmatrix} \mathbf{x}\_1\\ \mathbf{x}\_2 \end{bmatrix} + \begin{bmatrix} b\_0\\ 0 \end{bmatrix} u + \begin{bmatrix} 0\\ 1 \end{bmatrix} \dot{f},\\ y = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} \mathbf{x}\_1\\ \mathbf{x}\_2 \end{bmatrix}. \end{cases} \tag{10}$$

The corresponding continuous expansion state observer can be established as:

$$\begin{cases} \varepsilon = z\_1 - y\_1\\ \left[\begin{array}{cc} \dot{z}\_1\\ \dot{z}\_2 \end{array}\right] = \left[\begin{array}{cc} -\beta\_1 & 1\\ -\beta\_2 & 0 \end{array}\right] \left[\begin{array}{cc} z\_1\\ z\_2 \end{array}\right] + \left[\begin{array}{cc} b\_0 & \beta\_1\\ 0 & \beta\_2 \end{array}\right] \left[\begin{array}{cc} u\\ y \end{array}\right]. \tag{11}$$

In Equation (11), *z*1 and *z*2 are the state variables of the linear extended state observer, which can be adjusted using the difference between the state variables *z*1 and the system output *y*. By selecting the appropriate observer gain coefficients β1 and β2, the observer state variables can be used to observe the system output *y* and the total disturbance *f*(*y*, *<sup>w</sup>*).

Estimate the total disturbance of the system through the expanded state variables, and compensate for the input side of the system:

$$
\mu = \frac{(-z\_2 + u\_0)}{b\_0}.\tag{12}
$$

If the estimation error of *z*2 to *f*(*y*, *w*) is excluded, Equation (9) is simplified to a pure integral link:

$$\dot{y} = (f(y, w) - z\_2) + u\_0 \approx u\_0. \tag{13}$$

The linear feedback control law uses proportional links:

$$
u\_0 = kp(\upsilon - z\_1). \tag{14}$$

In Equation (14), *kp* is the controller bandwidth and *v* is the given reference input. Pole assignment of the parameters of the observer and controller is performed using the bandwidth method [10]:

$$\begin{cases} \beta\_1 = 2\omega\_{0\prime} \\ \beta\_2 = \omega\_{0\prime}^2 \\ k\_p = \omega\_c. \end{cases} \tag{15}$$

where ω*c* is the controller gain and ω0 is the observer gain. The bandwidth ω0 of the observer is about 3 to 5 times that of the controller ω*c*.

### **3. Design and Performance Analysis of ADRC**

### *3.1. Design and Stability Proof of the Improved State Observer*

The traditional observer estimates the internal state variables of the system using the difference *e* between the estimated value *z*1 of the system output and the output *y* of the system. In Sun [25], it is pointed out that the observer should first track the output *y* of the system with *z*1, and then track the output *f* with *z*2. Before the tracking of the estimated value *z*1 to the system output *y* is completed, other state variables of the observer cannot complete the tracking of the corresponding state variables of the system. However, when the observation variable *z*1 can better track the output *y* of the system, the smaller error *e* makes it difficult to adjust other observation variables; therefore, we have to use a larger coefficient β2 to speed up the tracking of the observation variables to the real value. Meanwhile, to ensure the stability of the system, the value of β2 cannot be too large. In general, the gain coefficient of the observer β*i* increases by an order of magnitude, which is more serious in the higher-order ADRC. Yang et al. [26] mentioned that the extended observer could be improved by introducing the differential of the observation error, but the parameters of the controller were doubled and the parameters of the observer needed to be configured; therefore, it had to be further simplified.

From Equation (11), we can get:

$$\begin{cases} z\_1 = x\_1 + \mathfrak{e}\_r \\ z\_2 = \dot{z}\_1 + \beta\_1 \mathfrak{e} - b\_0 \mu. \end{cases} \tag{16}$$

The following equation can be obtained by sorting Equation (16):

$$\begin{cases} z\_1 = x\_1 + \mathfrak{e}\_\prime \\ z\_2 = x\_2 + \dot{\mathfrak{e}} + \beta\_1 \mathfrak{e}. \end{cases} \tag{17}$$

It can be seen from Equation (17) that the error between *z*2 and *x*2 is .*e* + β1*e*, and adjusting *z*2 by using it as a correction amount can speed up the convergence without significantly increasing the observer gain. Therefore, the classic LESO can be modified as follows:

$$\begin{cases} \varepsilon = z\_1 - \mathbf{x}\_1, \\ \dot{z}\_1 = z\_2 - \beta\_1 \varepsilon + b\_0 u\_\prime \\ \dot{z}\_2 = -\beta\_2 (\dot{\varepsilon} + \beta\_1 \varepsilon). \end{cases} \tag{18}$$

Equations (12), (14), and (18) form the improved Linear Active Disturbance Rejection Controller of Equation (8), whose structure is shown in Figure 6.

**Figure 6.** Structure of the improved controller.

### 3.1.1. Improved Stability Proof of LESO

Let *e*1 = *z*1 − *x*1 and *e*2 = *z*2 − *x*2. From Equations (10) and (18), we obtain:

$$\begin{cases}
\dot{\varepsilon}\_1 = \varepsilon\_2 - \beta\_1 \varepsilon\_1, \\
\dot{\varepsilon}\_2 = -\beta\_2 \varepsilon\_2 - w.
\end{cases} \tag{19}$$

Let *Y*1 = *e*1 and *Y*2 = *e*2 − β1*e*1 to obtain the error equation of the LESO system:

$$\begin{cases}
\dot{Y}\_1 = Y\_{2\prime} \\
\dot{Y}\_2 = -(\beta\_1 + \beta\_2)Y\_2 - \beta\_1\beta\_2Y\_1 - w.
\end{cases} \tag{20}$$

The characteristic equation of Equation (20) is:

$$
\lambda^2 + (\beta\_1 + \beta\_2)\lambda + \beta\_1\beta\_2 = 0.\tag{21}
$$

The necessary and sufficient conditions for the stability of the second-order system are β1 + β2 > 0 and β1β2 > 0. The zero solution (*<sup>e</sup>*1 = 0, *e*2 = 0) of the second-order constant-coefficient differential equation shown in Equation (20) is globally asymptotically stable because ω0 > 0 and ω*c* > 0 are stable.

When considering the disturbance *w*, the system has a steady-state error. Specify *w*0 = *const* > 0 when |*w*| ≤ *w*0 . When the system reaches a steady-state, then:

$$\begin{cases} \dot{Y}\_1 = Y\_2 = 0, \\ \dot{Y}\_2 = 0. \end{cases} \tag{22}$$

The steady-state error is calculated according to Equation (19):

$$\begin{cases} \left| \left| \boldsymbol{e}\_{1} \right| \leq \frac{\text{uv}\_{0}}{\beta\_{1} \beta\_{2}} \prime \\ \left| \left| \boldsymbol{e}\_{2} \right| \leq \frac{\text{uv}\_{0}}{\beta\_{2}} \right. \end{cases} \tag{23}$$

### 3.1.2. Observation Errors of the Classical First-Order LESO

Stability and error analyses are performed on the traditional first-order LESO represented by Equation (11). Let *Y*1 = *e*1 and *Y*2 = *e*2 − β1*e*1 be used to obtain the equation of the traditional LESO error system:

$$\begin{cases}
\dot{Y}\_1 = Y\_2, \\
\dot{Y}\_2 = -\beta\_1 Y\_2 - \beta\_2 Y\_1 - w.
\end{cases} \tag{24}$$

 The characteristic equation of Equation (24) is given as follows:

$$
\lambda^2 + \beta\_1 \lambda + \beta\_2 = 0.\tag{25}
$$

Using the Hurwitz theorem, the necessary and sufficient conditions for the stability of the second-order system are β1 > 0 and β2 > 0. The zero solution (*<sup>e</sup>*1 = 0, *e*2 = 0) of the second-order constant-coefficient differential equation shown in Equation (24) is globally asymptotically stable because ω0 > 0 and ω*c* > 0 are stable.

When considering the disturbance *w*, the system has a steady-state error. Specify *w*0 = *const* > 0 when |*w*|≤ *w*0 . When the system reaches a steady-state, then:

$$\begin{cases} \dot{Y}\_1 = Y\_2 = 0, \\ \dot{Y}\_2 = 0. \end{cases} \tag{26}$$

The steady-state error of the observer can be expressed as:

$$\begin{cases} \left| \left| e\_1 \right| \le \frac{\overline{\mu\_0}}{\overline{\beta\_2}}, \epsilon \\ \left| \left| e\_2 \right| \le \frac{\overline{\beta\_1} w\_0}{\overline{\beta\_2}}. \end{cases} \right. \tag{27}$$

According to the above analysis, the modified LESO shown in Equation (18) can exhibit a better dynamic regulation performance and a smaller steady-state observation error than the traditional LESO when the parameters β1 and β2 are the same. Compared with Equations (23) and (27), the improved LESO exhibits a higher observation accuracy than the traditional LESO when the observer and controller bandwidths are the same.

### *3.2. Performance Index Analysis of the Improved Linear ADRC*

### 3.2.1. Convergence and Estimation Error of the Improved LESO

The Laplace transform of Equation (18) can be used to obtain the transfer function of the observer:

$$\begin{cases} Z\_1(\mathbf{s}) = \frac{(\beta\_1 + \beta\_2)s + \beta\_1\beta\_2}{(s + \beta\_1)(s + \beta\_2)} Y(\mathbf{s}) + \frac{b\_0 s}{(s + \beta\_1)(s + \beta\_2)} \mathrm{l}I(\mathbf{s}),\\ Z\_2(\mathbf{s}) = \frac{\beta\_2 s}{s + \beta\_2} Y(\mathbf{s}) - \frac{b\_0 \beta\_2}{s + \beta\_2} \mathrm{l}I(\mathbf{s}). \end{cases} \tag{28}$$

Taking *e*1 = *z*1 − *y* and *e*2 = *z*2 − .*y* into account for analyzing a typical *y*, and *u* as the amplitude *K* step signal, then the steady-state error of LESO is given as:

$$\begin{cases} \mathcal{e}\_1 = \lim\_{s \to 0} E\_1(s) = 0, \\\mathcal{e}\_2 = \lim\_{s \to 0} E\_2(s) = 0. \end{cases} \tag{29}$$

Equation (29) shows that the improved LESO has a good convergence performance for realizing the invariant estimation of the system state variables and generalized disturbances. Further analyzing its dynamic process, the response of *z*1 under the step signal when *b*0 = 0 is as follows:

$$z\_1(s) = K(\frac{1}{s} + \frac{2}{(s + 2\omega\_0)(\omega\_0 - 2)} - \frac{\omega\_0}{(s + \omega\_0^2)(\omega\_0 - 2)}).\tag{30}$$

The time-domain response of *z*1 under the action of a step signal can be obtained using an inverse Laplace transform:

$$z\_1(t) = \begin{cases} K(1 + \frac{2\varepsilon^{-2u\omega\_0} - \omega\eta e^{-4t}\overline{0}}{\omega\eta - 2}) & \omega\eta\_0 \neq 2, \\\ K(1 + 4te^{-4t} - e^{-4t}) & \omega\eta\_0 = 2. \end{cases} \tag{31}$$

In Equation (30), for *t* > 0, the derivative of *t* and taking .*<sup>z</sup>*1(*t*) = 0 can produce the extreme point of *t*0:

$$t\_0 = \begin{cases} \frac{2(\ln a\_0 - \ln 2)}{\omega\_0^2 - 2\omega\_0} & a\_0 \neq 2, \\\frac{1}{2} & a\_0 = 2. \end{cases} \tag{32}$$

The extreme value of *z*1 is obtained by substituting the extreme value *t*0 into Equation (30):

$$z\_1(t\_0) = \begin{cases} K(1 + \frac{\frac{-4(\ln^{40} - \ln^2)}{\omega\_0 - 2} - \frac{2\omega\_0(\ln^{40} - \ln^2)}{\omega\_0 - 2}}{-a\nu\_0 - 2}) & a\_0 \neq 2, \\ K(1 + e^{-2}) \approx 1.135K & a\_0 = 2. \end{cases} \tag{33}$$

The trajectory of *<sup>z</sup>*1(*<sup>t</sup>*0) with the observer bandwidth value ω0 can be obtained via digital simulation. According to Figure 7, when ω0 = 2, the tracking overshoot of observer *<sup>z</sup>*1(*<sup>t</sup>*0) to *y* is the largest. At this time, the system overshoot is equal to the traditional LESO overshoot. The traditional LESO has a 13.5% overshoot at *t*0 = 2/<sup>ω</sup>0 and the amount of overshoot is independent of the value of the observer bandwidth ω0.

**Figure 7.** Relation curve between the system overshoot and observer bandwidth.

The overshoot of the improved LESO varies with the observer bandwidth ω0. The maximum overshoot of the observer state variable *z*1 is equal to that of the traditional observer. The corresponding speed of the system can be increased and the amount of overshoot is reduced by selecting a larger observer bandwidth ω0. Although the observer bandwidth ω0 is larger and the tracking speed is faster, it will lead to noise amplification. The ability of LESO to suppress noise needs to be analyzed.

### 3.2.2. Improved Disturbance Immunity Analysis of LESO

The closed-loop transfer function of the improved LADRC can be obtained by combining Equations (12), (14), and (18):

$$
\mu = \frac{1}{b\_0} G\_1(s) (k\_p v - H(s)y). \tag{34}
$$

The transfer functions of *<sup>G</sup>*1(*s*) and *<sup>H</sup>*(*s*) in Equation (34) are as follows:

$$\begin{array}{l} \mathcal{G}\_{1}(\mathbf{s}) = \frac{(s + \beta\_{1})(s + \beta\_{2})}{s^{2} + (\beta\_{1} + k\_{\mathcal{P}})\mathbf{s}},\\ H(\mathbf{s}) = \frac{\beta\_{2}s^{2} + (\beta\_{1}\beta\_{2} + \beta\_{1}k\_{\mathcal{P}} + \beta\_{2}k\_{\mathcal{P}})s + \beta\_{1}\beta\_{2}k\_{\mathcal{P}}}{(s + \beta\_{1})(s + \beta\_{2})}. \end{array} \tag{35}$$

According to Equation (34), the diagram of the system structure shown in Figure 8 is obtained [24]:

**Figure 8.** Equivalent system structure diagram.

Now the effect of the observation noise δ0 at the output *y* of the system and the disturbance δ*c* at the output *u* of the controller for the improved LESO will be discussed. Based on Equation (28), the transfer function of the improved LADRC is:

$$\frac{z\_1}{\delta\_0} = \frac{(2\omega\_0 + \omega\_0^2)s + 2\omega\_0^3}{(s + 2\omega\_0)(s + \omega\_0^2)}.\tag{36}$$

Similarly, the transfer function of δ0 of the traditional LESO's observation noise can be obtained as follows:

$$\frac{z\_1}{\delta\_0} = \frac{2a\nu s + a\_0^2}{\left(s + a\_0\right)^2}.\tag{37}$$

The transfer function of the disturbance δ*c* at the output of the LESO controller can be improved according to:

$$\frac{z\_1}{\delta\_c} = \frac{b\_0 s}{(s + 2\omega\_0)(s + \omega\_0^2)}.\tag{38}$$

The transfer function of the disturbance δ*c* at the output of a traditional LESO controller is:

$$\frac{z\_1}{\delta\_c} = \frac{b\_0 s}{\left(s + w\_0\right)^2}.\tag{39}$$

Figure 9 shows the characteristic amplitude and phase–frequency curves for the improved and traditional LESOs. The bandwidth of the improved LESO was higher than that of the traditional LESO, and the phase lag of the intermediate frequency segmen<sup>t</sup> was improved. Unlike Figure 9, the improved LESO with the same observer bandwidth in Figure 10 is basically the same as the traditional LESO in the high frequency band, but it has better noise immunity in the low frequency band compared with the traditional LESO, and can more effectively suppress the interference at the input end.

**Figure 9.** Frequency-doman characteristic curves of the observed noise.

**Figure 10.** Frequency-domain characteristic curves of the input disturbance.

3.2.3. Immunity Analysis of the Improved Self-Disturbance Rejection Controller

According to Equation (10), the control object can be written as:

$$s\mathcal{Y}(\mathbf{s}) = F(\mathbf{s}) + b\eta \mathcal{U}(\mathbf{s}).\tag{40}$$

Combined with Figure 8, the closed-loop transfer function of the system is as follows:

$$Y(s) = \frac{k\_p}{s + k\_p} V(s) + \frac{s^2 + (\beta\_1 + k\_p)s}{(s + \beta\_1)(s + \beta\_2)(s + k\_p)} F(s). \tag{41}$$

The closed-loop transfer function of the system includes the tracking term and the disturbance term. If the state variable of the observer can be used to accurately estimate the total disturbance of the system, the closed-loop transfer function of the system is simplified to the first-order inertial link. At this time, it relates the corresponding speed of the system to the bandwidth of the controller, and the larger the bandwidth, the faster the system.

It can be seen from the closed-loop transfer function that the disturbance term impacts the observer and controller bandwidths. The same observer and controller bandwidths were selected for comparing the improved LESO with the traditional LESO. It can be seen from Figure 11 that under the same bandwidth, the immunity of the improved LESO in the middle- and low-frequency bands were better than that of the traditional LESO, and it improved the phase lag of the middle-frequency band.

**Figure 11.** Logarithmic phase–frequency characteristic curves of the disturbance term.

In particular, if the disturbance *f* is taken as the unit step signal, we can obtain the output response using Equation (41):

$$Y(s) = \frac{a}{(s + a\iota\_0)} + \frac{b}{(s + a\iota\_0^2)} + \frac{c}{(s + a\iota\_0)}\,\tag{42}$$

$$\begin{cases} a = \frac{a\eta + \omega\_{\!\!k}}{(\omega\_0^2 - \omega\_0)(\omega\_{\!\!k} - \omega\_0)}, \\\ b = \frac{2\omega\eta + \omega\_{\!\!k} - \omega\_0^2}{(\omega\_0 - \omega\_0^2)(\omega\_{\!\!k} - \omega\_0^2)}, \\\ c = \frac{2\omega\_0}{(\omega\_0 - \omega\_{\!\!k})(\omega\_0^2 - \omega\_{\!\!k})}. \end{cases} \tag{43}$$

The time domain for the anti-Laplace transformation to obtain system output can be expressed as:

$$y(t) = ae^{-a\nu\_0 t} + b\nu^{-\omega\_0^2 t} + c e^{-a\nu\_0 t}.\tag{44}$$

It is easy to find lim*<sup>t</sup>*→∞*y*(*t*) = 0, i.e., the steady-state output of the system is zero under the external step disturbance.

### **(a) Controller Stability with an Unknown Input Gain**

Considering only the influence of the input gain on the stability of the system, i.e., if *f* = (*b* − *b*0)*<sup>u</sup>*, then Equation (40) can be simplified to:

$$s\mathcal{Y}(\mathbf{s}) = b\mathcal{U}(\mathbf{s}).\tag{45}$$

The following equation can be obtained by combining Equations (26) and (30):

$$Y(s) = \frac{a\iota\_{\mathbb{H}}(s + 2a\iota\_{0})(s + a\_{0}^{2})}{a\_{3}s^{3} + a\_{2}s^{2} + a\_{1}s + a\_{0}}V(s). \tag{46}$$

In Equation (46), the coefficients are as follows: *a*3 = *b*0/*b*,*a*<sup>2</sup> = 2*a*0ω0 + *a*0ω*c* + ω20, *a*1 = <sup>2</sup>ω30 + 2<sup>ω</sup>0ω*c* + <sup>ω</sup>20<sup>ω</sup>*c*, and *a*0 = <sup>2</sup>ω30<sup>ω</sup>*c*. As ω0 and ω*c* are greater than zero, it is easy to see that *a*3,*a*2,*a*1 and *a*0 are all positive numbers. The necessary and sufficient condition for the stability of the Leonard qipat stability criterion (Equation (46)) is that all odd or even Hurwitz determinants are positive.

$$\begin{array}{c|cccc} \Delta\_3 &=& \begin{vmatrix} a\_2 & a\_0 & 0\\ a\_3 & a\_1 & 0\\ 0 & a\_2 & a\_0 \end{vmatrix} = a\_0(a\_1a\_2 - a\_0a\_3) \\ &= a\_0^2(4\omega\_0^4 + 2\omega\_0^3\omega\_\varepsilon + \omega\_\varepsilon^2\omega\_\varepsilon^2 + 4\omega\_0^2\omega\_\varepsilon + 2\omega\_0\omega\_\varepsilon^2) + (2\omega\_0^5 + \omega\_0^4\omega\_\varepsilon + 2\omega\_0^3\omega\_\varepsilon)a\_0 \end{array} \tag{47}$$

Since *b*, *b*0, ω0 and ω*c* are all positive numbers, Δ3 > 0 is true, i.e., the improved LADRC can be stable for any parameter greater than zero.

### **(b) Stability of the Controller when the System Parameters are Unknown**

Set the controlled object as the following:

$$y = \frac{b\_0}{s + k\_c} u\_\prime \tag{48}$$

where *kc* in Equation (48) is an unknown system parameter, and the closed-loop transfer function of the system is obtained by combining with Figure 8:

$$Y(s) = \frac{a\iota\_{\mathfrak{c}}(s+a\eta)(s+a\iota\_{0}^{2})(s+2a\eta)}{s^{4}+a\_{4}s^{3}+a\_{3}s^{2}+a\_{2}s+a\_{1}}V(\mathfrak{s}),\tag{49}$$

$$\begin{cases} a\_4 = \omega\_\emptyset^2 + 4\omega\_0 + \omega\_\varepsilon + k\_{\varepsilon\prime} \\ a\_3 = \omega\_0^2 \omega\_\varepsilon + 4\omega\_0^2 + 3\omega\_0^3 + 4k\_\varepsilon a\_0 + k\_\varepsilon a\_\varepsilon + 4\omega\_0 a\_\varepsilon \\ a\_2 = 4k\_\varepsilon \omega\_0^2 + 2\omega\_0^2 \omega\_\varepsilon + 3\omega\_0^3 \omega\_\varepsilon + 2\omega\_0^4 + 2k\_\varepsilon a\_0 \omega\_\varepsilon \\ a\_1 = 2\omega\_0^4 \omega\_\varepsilon. \end{cases} \tag{50}$$

The necessary and sufficient conditions for the stability of Equation (49) are:

$$a k\_c^3 + b k\_c^2 + c k\_c + d \gg 0,\tag{51}$$

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩*a* = <sup>4</sup>ω50<sup>ω</sup>*c*(<sup>4</sup><sup>ω</sup><sup>0</sup> + <sup>ω</sup>*c*)(<sup>2</sup><sup>ω</sup><sup>0</sup> + <sup>ω</sup>*c*), *b* = (<sup>72</sup>ω90<sup>ω</sup>*c* + <sup>68</sup>ω80ω2*c* + <sup>128</sup>ω80<sup>ω</sup>*c* + <sup>14</sup>ω70ω3*c* + <sup>160</sup>ω70ω2*c* + <sup>52</sup>ω60ω3*c* + <sup>4</sup>ω50ω4*c* , *c* = <sup>52</sup>ω110 ω*c* + <sup>62</sup>ω100 ω2*c* + <sup>176</sup>ω100 ω*c* + <sup>16</sup>ω90ω3*c* + <sup>252</sup>ω90ω2*c* + <sup>128</sup>ω90<sup>ω</sup>*c*<sup>+</sup> <sup>104</sup>ω80ω3*c* + <sup>272</sup>ω80ω2*c* + <sup>10</sup>ω70ω4*c* + <sup>144</sup>ω70ω3*c* + <sup>20</sup>ω60ω4*c* , *d* = <sup>12</sup>ω130 ω*c* + <sup>18</sup>ω120 ω2*c* + <sup>56</sup>ω120 ω*c* + <sup>6</sup>ω110 ω3*c* + <sup>96</sup>ω110 ω2*c* + <sup>64</sup>ω110 <sup>ω</sup>*c*+ <sup>48</sup>ω100 ω3*c* + <sup>160</sup>ω100 ω2*c* + <sup>6</sup>ω90ω4*c* + <sup>124</sup>ω90ω3*c* + <sup>64</sup>ω90ω2*c* + <sup>24</sup>ω80ω4*c*<sup>+</sup> <sup>72</sup>ω80ω3*c* + <sup>16</sup>ω70ω4*c* . (52)

If the roots of Equation (51) are *kc*1, *kc*2, and *kc*3 (*kc*1 < *kc*2 < *kc*3), the conditions of system stability are *kc*1 < *kc* < *kc*2 or *kc* > *kc*3.

According to the digital simulation results, the equation had a pair of conjugate complex roots and a real root. Figure 12 shows the boundary curve for ensuring the system stability when ω0 = 30 and ω0 ∈ [0, <sup>60</sup>], and Figure 13 shows the boundary curve for ensuring the system stability when ω0 = 30 and ω*c* ∈ [0, <sup>60</sup>]. From Figures 12 and 13, it can be seen that the stability region of the system increased with an increase of the observer bandwidth ω0 and the controller bandwidth ω*c*

**Figure 12.** System stability region under different observer bandwidths ω0.

**Figure 13.** System stability region under different controller bandwidth ω*c*.
